Question : Find the average of even numbers from 12 to 1598
Correct Answer 805
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1598
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1598 are
12, 14, 16, . . . . 1598
After observing the above list of the even numbers from 12 to 1598 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1598 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1598
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1598
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the even numbers from 12 to 1598
= 12 + 1598/2
= 1610/2 = 805
Thus, the average of the even numbers from 12 to 1598 = 805 Answer
Method (2) to find the average of the even numbers from 12 to 1598
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1598 are
12, 14, 16, . . . . 1598
The even numbers from 12 to 1598 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1598
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the even numbers from 12 to 1598
1598 = 12 + (n – 1) × 2
⇒ 1598 = 12 + 2 n – 2
⇒ 1598 = 12 – 2 + 2 n
⇒ 1598 = 10 + 2 n
After transposing 10 to LHS
⇒ 1598 – 10 = 2 n
⇒ 1588 = 2 n
After rearranging the above expression
⇒ 2 n = 1588
After transposing 2 to RHS
⇒ n = 1588/2
⇒ n = 794
Thus, the number of terms of even numbers from 12 to 1598 = 794
This means 1598 is the 794th term.
Finding the sum of the given even numbers from 12 to 1598
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given even numbers from 12 to 1598
= 794/2 (12 + 1598)
= 794/2 × 1610
= 794 × 1610/2
= 1278340/2 = 639170
Thus, the sum of all terms of the given even numbers from 12 to 1598 = 639170
And, the total number of terms = 794
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 12 to 1598
= 639170/794 = 805
Thus, the average of the given even numbers from 12 to 1598 = 805 Answer
Similar Questions
(1) What is the average of the first 893 even numbers?
(2) Find the average of the first 4925 even numbers.
(3) Find the average of odd numbers from 9 to 883
(4) Find the average of odd numbers from 15 to 473
(5) Find the average of the first 1502 odd numbers.
(6) Find the average of odd numbers from 9 to 679
(7) Find the average of even numbers from 8 to 830
(8) Find the average of odd numbers from 13 to 847
(9) Find the average of odd numbers from 13 to 237
(10) Find the average of odd numbers from 13 to 241